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arXiv:2211.00655 · Physical Review Letters

Coherent-Error Threshold for Surface Codes from Majorana Delocalization

Florian Venn, Jan Behrends, Benjámin Béri

Published Aug 8, 2023• 40 citations
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Abstract

Statistical mechanics mappings provide key insights on quantum error correction. However, existing mappings assume incoherent noise, thus ignoring coherent errors due to, e.g., spurious gate rotations. We map the surface code with coherent errors, taken as $X$ or $Z$ rotations (replacing bit or phase flips), to a two-dimensional (2D) Ising model with complex couplings, and further to a 2D Majorana scattering network. Our mappings reveal both commonalities and qualitative differences in correcting coherent and incoherent errors. For both, the error-correcting phase maps, as we explicitly show by linking 2D networks to 1D fermions, to a ${\mathbb{Z}}_{2}$-nontrivial 2D insulator. However, beyond a rotation angle ${\ensuremath{\phi}}_{\mathrm{th}}$, instead of a ${\mathbb{Z}}_{2}$-trivial insulator as for incoherent errors, coherent errors map to a Majorana metal. This ${\ensuremath{\phi}}_{\mathrm{th}}$ is the theoretically achievable storage threshold. We numerically find ${\ensuremath{\phi}}_{\mathrm{th}}\ensuremath{\approx}0.14\ensuremath{\pi}$. The corresponding bit-flip rate ${\mathrm{sin}}^{2}({\ensuremath{\phi}}_{\mathrm{th}})\ensuremath{\approx}0.18$ exceeds the known incoherent threshold ${p}_{\mathrm{th}}\ensuremath{\approx}0.11$.

DOI
10.1103/physrevlett.131.060603

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We consider the combined effect of readout errors and coherent errors, i.e., deterministic phase rotations, on the surface code. We use a recently developed numerical approach, via a mapping of the physical qubits to Majorana fermions. We show how to use this approach in the presence of readout errors, treated on the phenomenological level: perfect projective measurements with potentially incorrectly recorded outcomes, and multiple repeated measurement rounds. We find a threshold for this combination of errors, with an error rate close to the threshold of the corresponding incoherent error channel (random Pauli-Z and readout errors). The value of the threshold error rate, using the worst case fidelity as the measure of logical errors, is 2.6%. Below the threshold, scaling up the code leads to the rapid loss of coherence in the logical-level errors, but error rates that are greater than those of the corresponding incoherent error channel. We also vary the coherent and readout error rates independently, and find that the surface code is more sensitive to coherent errors than to readout errors. Our work extends the recent results on coherent errors with perfect readout to the experimentally more realistic situation where readout errors also occur.
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2D compass codes are a family of quantum error-correcting codes that contain the Bacon-Shor codes, the XXX-Shor and ZZZ-Shor codes, and the rotated surface codes. Previous numerical results suggest that the surface code has a constant accuracy and coherence threshold under uniform coherent rotation. However, having analytical proof supporting a constant threshold is still an open problem. It is analytically proven that the toric code can exponentially suppress logical coherence in the code distance LLL. However, the current analytical lower bound on the threshold for the rotation angle θθθ is ∣sin⁡(θ)∣<1/L|\sin(θ)| < 1/L∣sin(θ)∣<1/L, which linearly vanishes in LLL instead of being constant. We show that this lower bound is achievable by the ZZZ-Shor code which does not have a threshold under stochastic noise. Compass codes provide a promising direction to improve on the previous bounds. We analytically determine thresholds for two new compass code families that provide upper and lower bounds to the rotated surface code's numerically established infidelity threshold. Furthermore, using a Majorana mode-based simulator, we use random families of compass codes to smoothly interpolate between the ZZZ-Shor codes and the XXX-Shor codes.